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Cometric Association Schemes

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### Cometric Association Schemes

Worcester Polytechnic Institute

USA

Geometric and Algebraic Combinatorics 4, Oisterwijk, Thursday 21 August 2008

Several Collaborators

- Jason Williford
- Misha Muzychuk
- Edwin van Dam
- Nick LeCompte (WPI student)
- Will Owens (WPI student)
- . . . and I’ve received valuable suggestions from many others.

Today’s Goals

- Survey the known examples
- Summarize the main results to date
- Explore the structure of imprimitive
Q-polynomial schemes, especially with

3 or 4 classes

- List some open problems, big and small

My Real Goals

- To make the next 45 minutes as pleasant as possible

My Real Goals

- To make the next 45 minutes as pleasant as possible (for both you and me)

My Real Goals

- To make the next 45 minutes as pleasant as possible (for both you and me)
- To not look too dumb

My Real Goals

- To make the next 45 minutes as pleasant as possible (for both you and me)
- To not look too dumb
- To get some smart people to work on these interesting problems

My Real Goals

- To make the next 45 minutes as pleasant as possible (for both you and me)
- To not look too dumb
- To get some smart people to work on these interesting problems
- To tell you as much as I reasonably can about the subject

My Real Goals

- To make the next 45 minutes as pleasant as possible (for both you and me)
- To not look too dumb
- To get some smart people to work on these interesting problems
- To tell you as much as I reasonably can about the subject
- To avoid typesetting math in PowerPoint

First, an Example

E8 Root Lattice

Krein Parameters in Disguise

The Polytope Definition

The Polytope Definition

The Polytope Definition

Inner product of two zonal polynomials only depends on distance between the two base points and the single-variable polynomials.

The Bose-Mesner Algebra

Orthogonality Relations

Polynomial Schemes

Delsarte (1973):

Some Natural Questions

Concerning cometric association schemes . . .

What do they look like?

- I don’t know
- The model I just showed you is my favorite definition so far

Balanced Set Condition

Terwilliger (1987):

Sources of Examples

- Q-polynomial distance-regular graphs (e.g., all those with classical parameters)
- Spherical designs / lattices
- Extremal codes and block designs
- Real mutually unbiased bases
- Sporadic groups (e.g., triality)
- linked systems of designs and geometries

What do the Imprimitive Cometric Schemes Look Like?

Duality and Imprimitivity

w=3 fibres of size r=2

w=2 fibres of size r=3

A familiar dual pair of association schemes

Duality and Imprimitivity

Another dual pair of complete multipartite schemes

Suzuki’s Theorem

H. Suzuki (1998):

3-Class Cometric Schemes

Edwin van Dam (1995)

Don Higman’sTriality Scheme

Donald Higman (1928-2006)

Hyperovals in PG(2,4)

This is a 4-class Q-antipodal association scheme

Real MUBs

Real MUBs

Four-Class Schemes from MOLS

A Construction of Wocjan and Beth (2005)

Four-Class Schemes from MOLS

A Construction of Wocjan and Beth (2005)

Linked System of Symmetric Designs

- 48 vertices, split into three classes of size 16
- Graph G1represents “incidence”, yielding a
square (16,6,2)-design between any two

Q-antipodal classes

- “linked”: the number of common neighbors in the third class of a point chosen from Class One and a point chosen from Class Two depends on only whether or not these are incident (1 and 3, resp.)

MUBs from Cameron-Seidel Scheme

- Muzychuk, Williford, WJM introduced the extended Q-bipartite double
- Applied to the subschemes of the Cameron-Seidel scheme, these are 4-class Q-bipartite, Q-antipodal schemes
- So we have the same schemes that Bannai and Bannai found from mutually unbiased bases

More Material in Beamer Format

Check time available

Shortest Vectors in Leech Lattice

- 196560 vectors in R24, all of squared length 8
- only 7 possible inner products: ±8, ±4, ±2, 0
- construct one graph for each inner product
- we obtain a 7-class cometric scheme which is Q-bipartite
- Krein array:
- {24, 23, 288/13, 150/7, 104/5, 81/4;
- 1, 24/13, 18/7, 16/5, 15/4, 24 }

One More Photo

Heather Lewis and …?

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